$55)$ Let $\backslash ($ A$[1..$ n$]\backslash )$ contain $\backslash ($ n $\backslash )$ positive distinct integers in the range $\backslash (\backslash$left$[1\backslash$ldots n$^\{3\}\backslash$right$]\backslash ).$ And let $\backslash ($ V $\backslash )$ be a target value. In this exercise, you are to decide which of three proposed codes solve the following problem: Present a pseudocode to compute the number of different coin collections that sum up to equal $\backslash ($ V $\backslash )$ when the collections are formed by using

$0,1,$ or $2$ copies of $\backslash ($ A$[1]$ ; $0,1\backslash ),$ or $2$ copies of $\backslash ($ A$[2]$ ; $0,1\backslash ),$ or $2$ copies of $\backslash ($ A$[3]$ ; $\backslash$cdots ; $0,1\backslash ),$ or $2$ copies of $\backslash ($ A$[$n$]\backslash ).$ In parts $\backslash ($ i $\backslash )),\backslash ($ i i $\backslash )),$ and $\backslash ($ i i i $\backslash )),$ three different candidate adaptations of the standard program below are proposed. At least one is correct, and at least one is incorrect.

For each part, state if its adaptation is correct or incorrect. IN ADDITION, explain why each adaptation is wrong or right. The explanation can be brief: it counts some coin collection$($s$)$ more than once because $....,$ or it fails to count some coin colection$($s$)$ because $...,$ or it works because $....$

The purpose of this problem is to enhance your ability to think though what is happening, and think about the timing of the executions and what might go wrong. It is not an exercise to "solve" via implementation. Indeed, it is better to think more, get the answer wrong, and learn from your mistake when you see the solution that get the write answer based on running codes.

Each proposed solution adapts a standard $"$BFS$-$style subset sum count the ways program," which is reproduced here. So count $\backslash ([$j$]\backslash )$ stores the number of coin collections whose coins have been found $($so far$)$ to add up to equal $\backslash ($ j $\backslash ).$ But it is more convenient to think of count as storing the many $($distinct$)$ coin collections instead of how many have been found.

For each part below, state if its adaptation is correct or incorrect. IN ADDITION, explain why each INCORRECT adaptation is wrong. Likewise, give a brief justification about why each answer that you believe to be correct is actually correct. The proposed modifications are described below.

$5\mathrm{.}1)$ Create the new coin value array $\backslash ($ B$[1.\mathrm{.}2$ n$]\backslash )$ where $\backslash ($ B$[2$ i$-1]\backslash )$ equals $\backslash ($ A$[$i$]\backslash ),$ and $\backslash ($ B$[2$ i$]\backslash )$ equals $\backslash (2$ A$[$i$]\backslash ),$ for $\backslash ($ i$=1,2,\backslash$cdots$,$ n $\backslash ).$

Run the program once for the coin values $\backslash (\backslash$mathrm$\{$B$\}[1.\mathrm{.}2\backslash$mathrm$\{$n$\}]\backslash ).$

Correct, or Incorrect, and Why it is correct or incorrect?

$5\mathrm{.}2)$ Run the program once, but include the changes:

insert a line $5\mathrm{.}5($i$.$e$.,$ between lines $5$ and $6),$ which reads: if $\backslash ($ j$>2$ C $\backslash )$ then count $\backslash ([$j$]\backslash$leftarrow $\backslash$operatorname$\{$count$\}[$j$]+\backslash$operatorname$\{$count$\}[$j$-2$ C$]\backslash )$ endif.

insert the line: increment count $\backslash ([2$ C$]\backslash )$ right after line $7,$ which says: increment $\backslash (\backslash$operatorname$\{$count$\}[$C$]\backslash ).$

Correct, or Incorrect, and Why it is correct or incorrect?

$5\mathrm{.}3)$ Run the program twice, but but include the following changes:

Omit the print statement in line $9$ for the first run.

For the second run, omit the initialization of count in line $1.$

For the second run, omit line $7,$ which says: increment Kcount $\backslash ([$C$]\backslash )$; the second run forces count $\backslash ([2$ C$]\backslash )$ to count two $\backslash ($ C $\backslash )-$coins.

Correct, or Incorrect, and Why it is correct or incorrect?